A concurrent approach to the periodic event scheduling problem

Borndörfer, Ralf; Lindner, Niels; Roth, Sarah

doi:10.1016/j.jrtpm.2019.100175

Cited by 11 publications

(14 citation statements)

References 7 publications

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“…6 on the computation of the others. The cyclomatic number, also known as feedback edge set number, is a common measure for the difficulty of PESP instances, as it counts the number of integral variables used in a cycle-based mixed integer programming formulation (Borndörfer et al, 2019). However, parameters such as treewidth and branchwidth are much smaller.…”

Section: Parameter Searchmentioning

confidence: 99%

“…In this subsection, we discuss fixedparameter tractability by cyclomatic number. The cyclomatic number is a common measure for the difficulty of PESP instances, as it counts the number of integral variables used in a cycle-based mixed integer programming formulation (Borndörfer et al, 2019).…”

Section: Cyclomatic Numbermentioning

confidence: 99%

“…Contracting vertices of degree 2 is a preprocessing technique that is applied to large PESP instances, for example, in Goerigk and Liebchen (2017) and Borndörfer et al (2019).…”

Section: Cyclomatic Numbermentioning

confidence: 99%

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An analysis of the parameterized complexity of periodic timetabling

Lindner

Reisch²

2022

J Sched

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Public transportation networks are typically operated with a periodic timetable. The periodic event scheduling problem (PESP) is the standard mathematical modeling tool for periodic timetabling. PESP is a computationally very challenging problem: For example, solving the instances of the benchmarking library PESPlib to optimality seems out of reach. Since PESP can be solved in linear time on trees, and the treewidth is a rather small graph parameter in the networks of the PESPlib, it is a natural question to ask whether there are polynomial-time algorithms for input networks of bounded treewidth, or even better, fixed-parameter tractable algorithms. We show that deciding the feasibility of a PESP instance is NP-hard even when the treewidth is 2, the branchwidth is 2, or the carvingwidth is 3. Analogous results hold for the optimization of reduced PESP instances, where the feasibility problem is trivial. Moreover, we show W[1]-hardness of the general feasibility problem with respect to treewidth, which means that we can most likely only accomplish pseudo-polynomial-time algorithms on input networks with bounded tree- or branchwidth. We present two such algorithms based on dynamic programming. We further analyze the parameterized complexity of PESP with bounded cyclomatic number, diameter, or vertex cover number. For event-activity networks with a special—but standard—structure, we give explicit and sharp bounds on the branchwidth in terms of the maximum degree and the carvingwidth of an underlying line network. Finally, we investigate several parameters on the smallest instance of the benchmarking library PESPlib.

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Section: Parameter Searchmentioning

confidence: 99%

Section: Cyclomatic Numbermentioning

confidence: 99%

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An analysis of the parameterized complexity of periodic timetabling

Lindner

Reisch²

2022

J Sched

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“…This paper's author extended [PS16] to a general divide-and-conquer approach and applied the cycle-base formulation with several heuristics to improve on their solutions. Recently, [BLR19] combine Modulo Simplex, SAT, IP approaches and heuristics to deliver the current best solutions for PESPlib.…”

Section: Definition 2 ([Su89]mentioning

confidence: 99%

Finding Robust Periodic Timetables by Integrating Delay Management

Pätzold¹

2019

Preprint

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This paper defines and solves a mathematical model for finding robust periodic timetables by proposing an extension of the Periodic Event Scheduling Problem (PESP). In order to model delayed and not nominal travel times already in the timetabling step, we integrate delay management into the periodic timetabling problem. After revisiting both (PESP) and delay management individually, we introduce a periodic delay management model capable of evaluating periodic timetables with respect to delay resistance. Having introduced periodic delay management, we define the Robust Periodic Timetabling problem (RPT). Due to the high complexity of (RPT) we propose two different simplifications of the problem and introduce solution algorithms for both of them. These solution algorithms are tested against timetables found by standard procedures for periodic timetabling with respect to their delay-resistance. The computational results show that our algorithms yield timetables which can cope better with occurring delays, even on large-scale datasets and with low computational effort.

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“…However, a plenty of heuristic algorithms are available, connecting PESP with a zoo of well-known problems and techniques in combinatorial optimization, e.g., simplex algorithms [7,27], maximum cuts [19], matchings [29], and Boolean satisfiability [8,23]. The prevailing approach to solve PESP instances exactly is mixed-integer programming [3,15]. For example, the subway network of Berlin has been optimized by solving such a mixed-integer programming model [16].…”

Section: Introductionmentioning

confidence: 99%

The tropical and zonotopal geometry of periodic timetables

Bortoletto¹,

Lindner²,

Masing³

2022

Preprint

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The Periodic Event Scheduling Problem (PESP) is the standard mathematical tool for optimizing periodic timetabling problems in public transport. A solution to PESP consists of three parts: a periodic timetable, a periodic tension, and integer periodic offset values. While the space of periodic tension has received much attention in the past, we explore geometric properties of the other two components, establishing novel connections between periodic timetabling and discrete geometry. Firstly, we study the space of feasible periodic timetables, and decompose it into polytropes, i.e., polytopes that are convex both classically and in the sense of tropical geometry. We then study this decomposition and use it to outline a new heuristic for PESP, based on the tropical neighbourhood of the polytropes. Secondly, we recognize that the space of fractional cycle offsets is in fact a zonotope. We relate its zonotopal tilings back to the hyperrectangle of fractional periodic tensions and to the tropical neighbourhood of the periodic timetable space. To conclude we also use this new understanding to give tight lower bounds on the minimum width of an integral cycle basis.

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A concurrent approach to the periodic event scheduling problem

Cited by 11 publications

References 7 publications

An analysis of the parameterized complexity of periodic timetabling

An analysis of the parameterized complexity of periodic timetabling

Finding Robust Periodic Timetables by Integrating Delay Management

The tropical and zonotopal geometry of periodic timetables

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